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Our recent arXiv preprints and published papers on the solution of theRiemann-Lanczos and Weyl-Lanczos problems have brought our attention on theimportance of revisiting the algebraic structure of the Bianchi identities inRiemannian geometry. We also discovered in the meantime that, in our first bookof 1978, we had already used a new way for studying the various compatibilityconditions (CC) of an operator that may not be necessarily formally integrable(FI) in order to construct canonical formally exact differential sequences onthe jet level. The purpose of this paper is to prove that the combination ofthese two facts clearly shows the specific importance of the Spencer operatorand the Spencer $\delta$-cohomology, totally absent from mathematical physicstoday. The results obtained are unavoidable because they only depend onelementary combinatorics and diagram chasing. They also provide for the firsttime the purely intrinsic interpretation of the respective numbers ofsuccessive first, second, third and higher order generating CC. However, ifthey of course agree with the linearized Killing operator over the Minkowskimetric, they largely disagree with recent publications on the respectivenumbers of generating CC for the linearized Killing operator over theSchwarzschild and Kerr metrics. Many similar examples are illustrating thesenew techniques, providing in particular the only symbol existing in theliterature which is 2-ayclic witout being of finite type, contrary to theconformal situation.

Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules